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Miss Battaglia AP Calculus

7-2 Volume: The Disk Method Objective: Find volume of a solid revolution using the disk method and washer method. Miss Battaglia AP Calculus. Rotate around the x-axis and find the volume of y= 1 / 2 x from 0 to 4. The Disk Method.

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Miss Battaglia AP Calculus

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  1. 7-2 Volume: The Disk MethodObjective: Find volume of a solid revolution using the disk method and washer method. Miss Battaglia AP Calculus

  2. Rotate around the x-axis and find the volume of y=1/2 x from 0 to 4.

  3. The Disk Method If a region in the plane is revolved about a line, the resulting solid is a solid of revolution, and the line is called the axis of revolution. The simplest such solid is a right circular cylinder or disk (formed by revolving a rectangle about an axis adjacent to one side of the rectangle. Volume of disk = (area of disk)(width of disk) = πR2Δx ΔV = πR2Δx

  4. The approximation of the volume of a solid becomes better as ||Δ||  0 (n  ∞). Volume of a solid =

  5. The Disk Method To find the volume of a solid of revolution with the disk method, use one of the following: Horizontal Axis of RevolutionVertical Axis of Revolution

  6. Using the Disk Method Find the volume of a solid formed by revolving the region bounded by the graph of and the x-axis (0 < x < π) about the x-axis.

  7. Revolving About a Line that is Not a Coordinate Axis Find the volume of a solid formed by revolving the region bounded by and g(x)=1 about the line y=1.

  8. Try these! 1. y = - x + 1 revolved about the x-axis from 0 to 1 2. y = 4 – x2 revolved about the x-axis from 0 to 2 3. revolved about the x-axis from 1 to 4 4. revolved about the x-axis from 0 to 3 5. y = 2 and from -3 to 3 6. revolved about the x-axis from 0 to 4 7. , y = 0 from 0 to 3 8. y = 2x2, y=0, from 0 to 2

  9. The Washer Method The disk method can be extended to cover solids of revolution with holes by replacing the representative disk with a washer.

  10. Using the Washer Method Find the volume of the solid formed by revolving the region bounded by the graph of and about the x-axis.

  11. Integrating with Respect to y, Two-Integral Case Find the volume of a solid formed by revolving the region bounded by the graph of y=x2+1, y=0, x=0, and x=1 about the y-axis.

  12. Manufacturing A manufacturer drills a hole in the middle of a metal sphere of radius 5 in. The hole has a radius of 3 in. What is the volume of the resulting metal ring?

  13. Classwork/Homework • Read 7.2 Page 465 #1-9 odd, 11, 13

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